First, we consider the uncertainties of plasma density in the left half tube, with \(n_{ion,L}=n_{ele,L}=\xi \) .
The numerical expected solutions of macroscopic variables under different reference gyroradius at \(t = 0.1\)  are shown in Fig. REF .
It is known that the evolution of flow and electromagnetic fields can be well described by MHD equations when \(r_g\)  is small.
At \(r_g = 0.003\) , besides the typical wave structures in the classic Sod problem, a compound wave emerges around the tube center.
As illustrated by Brio and Wu [1]}, with the nonconvexity of ideal MHD equation, this wave structure is induced from initial magnetic discontinuity.
From Fig. REF (a)-(c), we see the current scheme from the BGK-Maxwell system provides equivalent numerical solutions as ideal MHD equations,
which identifies the asymptotic preserving (AP) property of the current scheme in the MHD limit.
